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arXiv · 2501.04935

Geodesic Variational Bayes for Multiway Covariances

Quinn Simonis·Martin T. Wells
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Abstract

This article explores the optimization of variational approximations for posterior covariances of Gaussian multiway arrays. To achieve this, we establish a natural differential geometric optimization framework on the space using the pullback of the affine-invariant metric. In the case of a truly separable covariance, we demonstrate a joint approximation in the multiway space outperforms a mean-field approximation in optimization efficiency and provides a superior approximation to an unstructured Inverse-Wishart posterior under the average Mahalanobis distance of the data while maintaining a multiway interpretation. We moreover establish efficient expressions for the Euclidean and Riemannian gradients in both cases of the joint and mean-field approximation. We end with an analysis of commodity trade data.

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Quinn Simonis, Martin T. Wells. 2025-01-09. Geodesic Variational Bayes for Multiway Covariances. https://arxiv.org/abs/2501.04935

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